CF1765C.Card Guessing

Consider a deck of cards. Each card has one of $4$ suits, and there are exactly $n$ cards for each suit — so, the total number of cards in the deck is $4n$. The deck is shuffled randomly so that each of $(4n)!$ possible orders of cards in the deck has the same probability of being the result of shuffling. Let $c_i$ be the $i$-th card of the deck (from top to bottom).

Monocarp starts drawing the cards from the deck one by one. Before drawing a card, he tries to guess its suit. Monocarp remembers the suits of the $k$ last cards, and his guess is the suit that appeared the least often among the last $k$ cards he has drawn. So, while drawing the $i$-th card, Monocarp guesses that its suit is the suit that appears the minimum number of times among the cards $c_{i-k}, c_{i-k+1}, \dots, c_{i-1}$ (if $i \le k$, Monocarp considers all previously drawn cards, that is, the cards $c_1, c_2, \dots, c_{i-1}$). If there are multiple suits that appeared the minimum number of times among the previous cards Monocarp remembers, he chooses a random suit out of those for his guess (all suits that appeared the minimum number of times have the same probability of being chosen).

After making a guess, Monocarp draws a card and compares its suit to his guess. If they match, then his guess was correct; otherwise it was incorrect.

Your task is to calculate the expected number of correct guesses Monocarp makes after drawing all $4n$ cards from the deck.

考虑一副扑克牌。每张牌有 $4$ 种花色之一，且每种花色恰好有 $n$ 张牌——因此，整副牌共有 $4n$ 张牌。这副牌被随机洗牌，使得 $(4n)!$ 种可能的牌序中每一种出现的概率均相等。令 $c_i$ 表示牌堆中自上而下的第 $i$ 张牌。

Monocarp 开始依次从牌堆中抽牌。在抽出一张牌之前，他尝试猜测该牌的花色。Monocarp 会记住最近抽到的 $k$ 张牌的花色，其猜测依据是：在最近抽到的 $k$ 张牌中，出现次数最少的花色。具体而言，在抽出第 $i$ 张牌时，Monocarp 猜测其花色为牌 $c_{i-k}, c_{i-k+1}, \dots, c_{i-1}$ 中出现次数最少的花色（若 $i \le k$，则 Monocarp 考虑所有此前已抽出的牌，即牌 $c_1, c_2, \dots, c_{i-1}$）。若在 Monocarp 所记住的前几张牌中，有多个花色出现次数同为最小值，则他从中随机选择一个作为猜测（即所有出现次数最少的花色被选中的概率相等）。

做出猜测后，Monocarp 抽出一张牌，并将其实际花色与自己的猜测进行比对。若二者一致，则此次猜测正确；否则为错误。

你的任务是计算 Monocarp 抽完全部 $4n$ 张牌后，所作猜测中正确的次数的期望值。

The first (and only) line contains two integers $n$ ($1 \le n \le 500$) and $k$ ($1 \le k \le 4 \cdot n$).

第一行（也是唯一一行）包含两个整数 $n$（$1 \le n \le 500$）和 $k$（$1 \le k \le 4 \cdot n$）。

Let the expected value you have to calculate be an irreducible fraction $\dfrac{x}{y}$. Print one integer — the value of $x \cdot y^{-1} \bmod 998244353$, where $y^{-1}$ is the inverse to $y$ (i. e. an integer such that $y \cdot y^{-1} \bmod 998244353 = 1$).

设你需要计算的期望值为一个既约分数 $\dfrac{x}{y}$。输出一个整数 —— $x \cdot y^{-1} \bmod 998244353$ 的值，其中 $y^{-1}$ 是 $y$ 的模逆元（即满足 $y \cdot y^{-1} \bmod 998244353 = 1$ 的整数）。